Numerical solution of the matrix equation X — A\bar XB = C in the self-adjoint case

The numerical algorithm for solving the equation X $A\bar XB$ = C proposed in an earlier publication is now modified for the situation where this equation can be regarded as a self-adjoint one. The economy in the computational time and work achieved through these modifications is illustrated by numerical results.     

The Z \to c\bar c \to \gamma \gamma *, Z \to b\bar b \to \gamma \gamma * triangle diagrams and the Z → γψ, Z → γγ decays

We expounded an approach for studying the Z ?? ??? and Z ?? ???? decay based on the sum rules for the $Z \to c\bar c \to \gamma \gamma *$ and $Z \to b\bar b \to \gamma \gamma *$ amplitudes and their derivatives. We calculate the branching ratios of the Z ?? ??? and Z ?? ???? decays under different suppositions about the saturation of the sum rules. We find the lower bounds of ?? _?? BR(Z ?? ???) = 1.95 · 10 ^?7 and ?? _?? BR(Z ?? ????) = 7.23 · 10^?7 and discuss deviations from the lower bounds including the possibility of BR[Z ?? ??J/??(1S)] ?? BR[Z ?? ????(1S)] ?? 10 ^?6 , which is probably measurable at the LHC. Moreover, we calculate the angle distributions in the Z ?? ??? and Z ?? ???? decays.     

Limits on an extension of Carleson’s\bar \partial - Theorem

We study a theorem essentially due to Carleson about solving the $\bar \partial - equation$ on the unit disk. We show that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, our main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. We exhibit a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to ∞. Since Carleson’s $\bar \partial - theorem$ depends on the notion of a Carleson measure, we also discuss Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.     

Positive\partial \bar \partial - closed currents and non-Kähler geometrycurrents and non-Kähler geometry

In this paper some new results on positive \(\partial \bar \partial - closed\) currents are applied to modifications \(f:\bar M \to M\) . The main result in this topic is that every smooth proper modification of a compact Kähler manifoldM is balanced. Moreover, under suitable hypotheses on the map, the Kähler degrees of \(\bar M\) corresponds to homological properties of the exceptional set of the modification. More examples ofp-Kähler manifolds are discussed in the last section of the paper.     

On the spectrum of the Cesàro operator C 1 on b\bar v_0 \cap \ell _\infty

In this paper we have investigated the spectrum of the Cesàro operator C ₁ which is regarded as an operator on the sequence space $b\bar v_0 \cap \ell _\infty $ the space of statistically null bounded variation sequences.     

Randregularität des\bar \partial -Problems für die Halbkugel in ℂn

Let f be a -closed (0, q+1)-form defined on the half ball B₊ in ⁿ, which is of class C^k on ¯B. We prove the existence with C^k-estimates of a solution u of the equation .

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Die Anregung zu dieser Arbeit verdanke ich auch Herrn Piotr Jakóbczak aus Krakau, der sich eine gewisse Zeit am Max-Planck-Institut für Mathematik in Bonn aufhielt. Während dieser Zeit untersuchten wir die C⁰-Abschätzungen für die Halbkugel in ⁿ     

矩阵方程X+A*X-nA=I的正定解

In this paper we give some sufficient conditions and some necessary conditions under which the matrix equation X A^*X^-nA=I has a positive definite solution. An iterative method which converges to a positive definite solution of this equation is constructed. And an error estimate formula on this iterative method is also derived.     

Hölder andL p -estimates for the\bar \partial -equation on non-smooth strictlyq-convex domainson non-smooth strictlyq-convex domains

A solution operator for the \(\bar \partial \) -equation on strictlyq-convex domains with nonsmooth boundary is constructed. It is proved that the solution satisfies optimal 1/2-Hölder andL ^p estimates.     

On the\bar \partial equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

For a large class of subharmonicφ, the equation is studied in . Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL ^p norms are derived as a corollary. A class of counterexamples, related to the failure of to be analytic-hypoelliptic on certain CR manifolds, is discussed. Communicated by Steven Krantz     

Optimal Lp estimates for the\bar \partial -equation on complex ellipsoids in ℂn-equation on complex ellipsoids in ℂn

In this paper we prove the best possible L^p estimates for the-equation on complex ellipsoids in ℂⁿ, and provide examples to show why they cannot be improved.     

The behavior of \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-\nulldelimiterspace} n}} for particular values of θ

Let ζ be a primitive q′-root of unity. We prove that the series $ \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-0em} n}} $ for θ ∈ Q converges if and only if θ = p/q with (p,q) = 1 and q′ ? p, and that there exists an uncountable set S of Liouville’s numbers such that the series does not converge when θ ∈ S.     

关于算子方程AXB=X

<正> 设N_1,N_2,X是Hilbert空间H上线性有界算子,并且N_1,N_2是正常的.如果N_1X=XN_2,则N_1X=XN_2.这是熟知的Putnam-Fuglede定理.它有许多推广(参见[2]).我们在文[2]中也曾讨论它的一些推广,特别,在[2]中我们讨论了如下的一种形式:设A,B,X为Hilbert空间H上线性有界算子,如果AXB=X,那么,在适当的条件下(例如对A,B,X加上某些限制),必有AXB=X.在文[2]中还给出了AXB=X,AXB=X同时成立的一个充要条件.本文是[2]的继续,继续讨论这类问题.     

Hölder estimates for the\bar \partial - equation in some domains of finite typein some domains of finite type

We obtain Hölder estimates for the $\bar \partial - equation$ on some domains of finite type in ?ⁿ using proper mapping techniques. The domains considered are domains of finite type in the sense of D’Angelo and are defined by local coordinate expressions satisfying certain algebraic geometric conditions which prevent the existence of complex analytic varieties in the boundary of the domain. Using a proper mapping which is given by the finite type condition and which carries all the information about the intrinsic geometry of the boundary, we transform the finite type points into strongly pseudoconvex ones. At these strongly pseudoconvex points we compute an explicit solution using the Henkin integral formula and we obtain estimates that we are able to pull back to the original domain. We achieve this by exploiting the branching behavior of the proper mapping. We also construct some biholomorphic numerical invariants associated with some of the domains under consideration.     

Unlösbarkeit der Gleichung $\alpha x^n + \beta y^n = z^n$ und Gleichverteilung von $\root n \of{\alpha x^n + \beta y^n}$

We show that the unsolvability of the Diophantine equation is equivalent to a good uniform distribution of the set . The proof depends on the asymptotic evaluation of the Gauss sum .     

一类复映射$z\leftarrow e^{i\phi}(\bar{z})^\alpha+c~ (\alpha<0)$的广义M集

本文分析了一类复映射$z \leftarrow e^{i\phi }(\bar {z})^\alpha +c\{\alpha < 0,\phi \in [0,2\pi)\}$的临界点的性质,给出了广义Mandelbrot集 (简称广义M集)的定义,并构造出一系列广义M集.利用复变函数理论和计算机制图相结合的实验数学的方法,本文对广义M集的结构和演化进行了研究,结果表明: 1). 广义M集的几何结构依赖于参数$\alpha$, $R$和$\phi$; 2). 整数阶广义M集具有对称性和分形特征; 3). 小     

关于求解矩阵方程I=X+A~HX~(-1)A的简单迭代

1 引言 设A为m×m方阵,I为m阶单位阵,考虑关于X的非线性矩阵方程 I=X+A~HX~(-1)A的Hermite正定解问题。这是特殊的离散代数Riccati方程,在一定条件下与离散代数Riccati方程数学等价。由于离散代数Riccati方程还缺乏普遍有效的数值解法,因此研究(1.1)的数值处理就十分重要。最近,Engwerda等学者研究了c1)、c2)方程(1.1)可解的充分必要条件、最大解和最小解的存在唯一性,还提出如下简单迭代 X_o=I,X_(n+1)=I-A~HX_n~(-1)A,n=0,1,….(1.2) 证明了{X_n}_(n=0)~∞收敛于(1.1)的极大解X_L.这项研究为数值求解(1.1)提供了可能.本文研究下述三方面问题.首先是(1.2)的误差估计,它同时也是迭代过程(1.2)的收敛速度估计.然后给出一种执行格式.由于(1.2)每迭代一步要计算一个m阶方阵的逆矩阵,计算量很大,因而提出有效的执行格式是必要的.最后研究极大解X_L的扰动定理. 若不特别说明,以下的记号都是常规的,例如可参阅[3]. 2 误差估计 令A的数值半径为ω(A).Engwerda和Ran证明了下列结果:设A可逆,那么(1.1)存在对称正定解的充要条件为ω(A)≤1/2;若(1.1)有对称正定解则有唯一的最大解X_L;若(1.1)有对称正定解,则(1.2)产生的矩阵序列{X_n}收敛到X_L,且收敛过程是单调下降的.     

Multiple Fourier sums and ϕ-strong means of their deviations on the classes of $$\bar \psi $$ -differentiable functions of many variables

We present the results concerning the approximation of -differentiable functions of many variables by rectangular Fourier sums in uniform and integral metrics and establish estimates for the ϕ-strong means of their deviations in terms of the best approximations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1075–1093, August, 2007.